Coordinate Systems for Astronomy or: How to get …avntraining.hartrao.ac.za/images/coordinate_systems_for...Coordinate Systems for Astronomy or: How to get your telescope to observe

Coordinate Systems for Astronomy

or: How to get your telescope to observe the right object

Definitions - Poles, Equator, Meridians, Parallels

The rotation of the Earth on its axis lets us define a coordinate system for the surface of the Earth.

The two points where the rotation axis meets the surface of the Earth are known as the north

geographic pole and the south geographic pole (Figure 1).

The equator is the great circle perpendicular to the rotation axis and lying half-way between the

poles.

Great circles which pass through the two poles are known as meridians.

Small circles which lie parallel to the equator are known as parallels.

Question: What other poles does the Earth have, that could be confused with the geographic poles?

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Figure 1: Basic definitions for the Earth

Coordinates on the Earth – Longitude, Latitude

The longitude of a point on the Earth’s surface is measured east or west along the equator. Its value

is the angular distance between the local meridian passing through the point and the Greenwich

Meridian, which passes through the Royal Greenwich Observatory in London (Figure 2).

Because the Earth is rotating on its own axis it is possible to express longitude in time units as well

as angular units. The Earth rotates through:

• 360° in 24 hours (of Sidereal Time – see later), or

• 15° of longitude in 1 hour, or

• 1° of longitude in 4 minutes, or

• 1' of longitude in 4 seconds, or

• 1" of longitude in 1/15 second.

The latitude of a point is the angular distance north or south of the equator, measured along the

meridian passing through the point. It is measured in degrees.

Question: What is the approximate longitude and latitude of Johannesburg?

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Figure 2: Earth Longitude and Latitude

Horizon Coordinates – Azimuth, Altitude / Elevation, Zenith Angle

The horizon coordinate system is also called the topocentric or alt-az or az-el system.

It is the simplest coordinate system as it is based on the observer's horizon. The celestial

hemisphere viewed by an observer on the Earth is shown in Figure 3.

The great circle through the zenith and the north celestial pole cuts the horizon at the north point

(N) and the south point (S). It is called the zenith meridian.

Altitude (or elevation) is the angle measured along a vertical circle from the horizon. It is

measured in degrees.

An alternative to altitude is the zenith angle, z = 90° - altitude.

Azimuth is generally defined as the angle between the vertical through the north point and the

vertical through the star, measured eastwards from the north along the horizon from 0° to 360°.

This definition applies to observers in both the northern and the southern hemispheres.

Question: How close to zenith does the Sun come in Johannesburg? When does this occur?

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Figure 3: Horizon coordinates for an observer

Celestial Equatorial Coordinates - Right Ascension, Declination

Because the altitude and azimuth of a star are different for different places on the Earth, the horizon

coordinate system is not useful for cataloguing celestial positions. A more useful coordinate

system for locating celestial objects is one based on the celestial equator and the celestial poles.

This is defined in a similar manner to latitude and longitude on the surface of the Earth. In this

system, known as the equatorial coordinate system, the analogue of latitude is the declination.

The declination (Dec. or δ) of a star is its angular distance in degrees measured from the celestial

equator along the meridian through the star. It is measured north and south of the celestial equator.

It ranges from 0° at the celestial equator to +90° at the north celestial pole and -90° at the south

celestial pole.

Right Ascension (R.A. or α) is the equivalent of longitude on the Earth. The zero point chosen on

the celestial sphere is the point at which the Sun crosses the equator, on about March 21. This is

called “The first point in Ares, γ”. It is the northern hemisphere vernal (spring) equinox, and the

southern hemisphere autumnal equinox.

The Right Ascension of an object is the angle between the meridians through the first point in Ares

and the celestial object. It is measured in units of time, from 0h to 24h along the celestial equator

eastwards (counter-clockwise) from the first point of Aries.

Figure 4 also shows the ecliptic, the path along which the Sun appears to move in the sky. Ecliptic

Coordinates are used for the positions of solar system objects, particularly the planets. Ecliptic

Longitude and Latitude are measured in degrees.

Question: Why is the ecliptic tilted, and by how much is it tilted?

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Figure 4: Celestial Equatorial coordinates

Precession – Epoch, Precession

Having settled on Right Ascension and Declination as the natural coordinate system for most

celestial objects, we find that the RA and Dec of celestial objects very inconveniently change with

time. The reason for this is that the direction in which the Earth’s axis points is changing, owing to

the tidal pull on the not quite spherical Earth. This effect is called Precession (Figure 5).

As a result, an Epoch must be stated in respect of all RA, Dec coordinates. Standard epochs for

catalogues of objects were B1900, B1950 and J2000. Here B standards for Besselian and J for

Julian, and describe subtle changes of definition.

To get the current position of an object in order to observe it, its coordinates must be precessed

from the catalogue epoch to the date and time of observation.

Generic Equatorial Coordinates plotted on a rectangular grid are shown in Figure 6.

Figure 6: RA, Dec coordinate system

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Figure 5: Precession of the Earth's axis

Galactic Coordinates – Longitude, Latitude

For objects in the Milky Way, the Galactic coordinate system is also used. The direction of zero

longitude and latitude is towards the centre of the Milky Way. This coordinate system does not vary

with time, unlike RA, Dec. As an early attempt at defining the origin of the Galactic Coordinates

was found later to be inaccurate, a second zero point was determined, and in this system Galactic

longitude, latitude are called Lii, Bii (Figure 7).

The centre of the Milky Way, ~7000 parsecs away towards Lii = 0o, Bii = 0o, lies in Sagittarius, at

J2000 RA 17h 45m 37s, Dec -28o 56' 10''. It passes over South Africa once per day.

The Galactic anti-centre direction, at Lii = 180o, Bii = 0o, lies north of the well known constellation

of Orion, at J2000 RA 5h 45m 37s, Dec 28o 56' 10'' Figure 8).

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Figure 7: Galactic Coordinates

Figure 8: Galactic coordinates mapped onto B1950 Equatorial coordinates

Radio emission from the sky is shown above in equatorial coordinates (Figure 9) and below in

Galactic coordinates (Figure 10).

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Figure 10: Radio sky at 2300MHz in Galactic coordinates (Jonas et al. 1998). Left one-third from Stockert telescope, Germany. Right two-thirds from HartRAO 26m telescope. Shown using the Mollweide equal area projection.

Figure 9: Radio sky at 408MHz in J2000 Equatorial coordinates (Haslam et al. 1982). Shown using rectangular projection, with RA in degrees.

Time – Universal Time, Sidereal Time, Julian Date

Universal Time comes in more than one form. Universal Time Coordinated (UTC) is also known

as Greenwich Mean Time (GMT). It is the time measured by a evenly running clock positioned

on the Greenwich Meridian.

The world is divided into Time Zones; 15 degrees change in longitude is equivalent to an hour's

change in time. South Africa is roughly 30 degrees east of Greenwich, and so South African

Standard Time (SAST) is set to be 2 hours ahead of UTC.

Our “year” is technically the Earth's “tropical period”. This is the time that elapses between two

alignments of its axis of rotation with the Sun, or 365.242 days. The Earth's orbital (sidereal)

period around the Sun is 365.256 days. The 0.014 day (= 20 minutes) difference is caused by

precession, which has a period of 25770 years (see above), as 365.25/25770 = 0.014 day/yr.

The Earth turns once on its axis in 23 hours 56 minutes and 04 seconds. This is called a sidereal

day. Stars rise at the same time each sidereal day.

However we set our watches to a length of day matching the mean solar day. The Earth has to turn

for an extra ~1o to get the Sun back to the same place in the sky, as it moves almost 1o per day in its

orbit around the Sun. The length of the mean solar day is the well known 24 hours.

To point our telescopes we have to make allowance for the actual, irregular rotation period of the

Earth. UT1 is the time scale used to measure this. Leap seconds are inserted in UTC at intervals to

keep UTC and UT1 within 0.9 of a second.

Measuring the changing length of day (UT1) is done by a global network of radio telescopes using

quasars far out in the universe as fixed radio beacons. The HartRAO 26m radio telescope is part of

this network.

A time scale divided into years with months of different lengths and numbers of days per month and

year is very awkward when dealing with time series of data. To get around this the Julian Date

(JD) was devised. This is the number of decimal days since 12h00 UT of January 1, 4713 BCE.

Modified Julian Days (MJD) are also used. They start at 0h00 UT, and are defined as

MJD = JD – 240000.5

The Julian Date on 2009 August 5 at 10h00 UT was 2455048.875, and the MJD was 55048.375.

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Figure 11: Solar day and sidereal day

Using Sidereal Time – Mean, Apparent, and Hour Angle

As with all longitude-like coordinates, a zero point has to be set for sidereal time. This is the

sidereal time at the Greenwich meridian, or Greenwich Mean Sidereal Time (GMST). It is the

elapsed time since the zenith meridian transit of the vernal equinox at the Greenwich meridian.

Local Mean Sidereal Time (LMST) is then the current Greenwich Mean Sidereal Time plus the

observer's longitude (East longitude is positive), converted from degrees to hours.

LMST[h] = GMST[h] + Longitude[o] / 15

A small correction (< 1.15 seconds of time) to this for nutation (18.6 year period wobble of the

Earth's pole) gives the Local Apparent Sidereal Time (LAST). This equals the Right Ascension

of all bodies currently crossing the observer's zenith meridian.

Observatories often display clocks showing the Local Sidereal Time to aid observers. For example,

see http://www.hartrao.ac.za/hart_lst.html .

The Hour Angle of an object is the angle of an object east or west of the observers zenith meridian.

Hour Angle = Local Sidereal Time – Right Ascension of Date

or simply:

HA = LST – RA

A negative Hour Angle translates as the time until an object crosses the zenith meridian, a positive

Hour Angle as the time since it has crossed the zenith meridian.

Question: The LST in Johannesburg is 22h 30m. Where in the sky is 3C48, one of the radio

galaxies used for flux calibration, whose coordinates are approximately RA 01h 34m +32o 54'?

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Refraction

Ground-based telescopes have to observe through the Earth's atmosphere. Refraction by the

atmosphere increases the apparent elevation angle of an object (Figure 12). The effect is largest for

objects close to the horizon. Refraction by the atmosphere is greater by about 8% for radio waves

compared to visible light. Refraction also depends on the height of the observatory above sea level

and on the water vapour content of the atmosphere.

A simple approximation to the radio refraction is:

Δz = (n - 1) x tan(z)

where Δz (radians) is the apparent increase in elevation angle at zenith angle z (radians)

n is the refractive index

n – 1 ~ 3 x 10-4 for radio waves at sea level

n – 1 ~ 2.5 x 10-4 for radio waves at Hartebeesthoek, at altitude 1400m.

Question: The horizon for the radio telescopes at Hartebeesthoek varies from 3o to 12o elevation,

as they are located in a valley. What is the approximate radio refraction in degrees at those two

elevation angles?

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Figure 12: Atmospheric refraction

Distances – Astronomical Unit, Parsec

For relatively small astronomical distances the Astronomical Unit (AU) can be used. This is

defined as the mean Sun-Earth distance, which is 149.6 x 106 km (Figure 13).

For large distances the Parsec (pc) is used. This is defined as being equal to the distance from the

Sun at which the mean Sun-Earth distance (1 AU) subtends an angle of 1 arc second (Figure 13).

One parsec equals 3.086 x 1013 km, or 3.26 light years.

Parallax

A consequence of the Earth's motion around the Sun is that nearby objects in space appear to move

forwards and backwards against the background of very distant objects, through parallax (Figure

14). The amount of parallactic movement is proportional to their distance, so the observed parallax

provides an absolute measure of their distance. Optically, this has long been used to measure the

distance to nearby stars.

Radio telescopes separated by thousands of kilometres can measure parallaxes of milliarcseconds

(mas). These networks of radio telescopes are now being used to measure the distances to star-

forming regions in the Milky Way, using the annual parallax of methanol and water vapour masers

embedded in the star-forming clouds. The aim is to accurately map the spiral arms of the Milky

Way. The HartRAO 26m telescope has been used to search for suitable masers for this project.

We have: d = 1 / p

where d = distance to source in parsecs

p = parallax angle of source in arcseconds.

Question: The methanol masers in the star-forming region G9.62+0.20 were found by Sharmila

Goedhart in her monitoring with the HartRAO 26m telescope to show regular flaring, the first ever

found in masers in a star-forming region. The parallax of the masers has been measured to be 0.194

mas. What is their distance?

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Figure 13: Definition of the Astronomical Unit and Parsec

Figure 14: Parallax measurement

Astronomical (Stellar) Aberration

Telescopes are not fixed in space, they are on a moving Earth. As the speed of light is finite, this

causes the apparent position of objects to shift. We have to point the telescope at the apparent

position of the object to allow for this stellar aberration (Figure 15).

q = angle to true position of star (D-B-C)

φ = angle (D-A-C) at which telescope must actually point to see the star owing to

v = horizontal motion to right of telescope owing to Earth velocity in orbit or rotation on own axis

h = length of telescope tube

c = speed of light

h / c = time taken for light to travel down telescope tube

v h / c = horizontal distance travelled by telescope while light passes down the tube

To solve for the aberration angle q – φ = B-D-E

in triangle BDE, sin (q - φ) = opposite / hypoteneuse = b / h

in triangle EAB, sin φ = b / (v h / c)

rearranging, b = (v h / c) sin φ

substituting for b: sin (q - φ) = v h / c . sin φ / h

so sin (q - φ) = v / c sin φ

hence as (q - φ) is small, the aberration angle q – φ ≈ sin (q – φ) = v / c sin φ

The maximum value possible for the aberration is v / c, the ratio of the two velocities. This is called

the constant of aberration.

Annual aberration was used by Bradley in 1727 to measure the speed of light.

Question: What is the constant of aberration in units of arc seconds for (a) annual aberration,

caused by the Earth orbiting the Sun, and (b) for daily aberration, for the Earth turning on its own

axis?

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Figure 15: Stellar Aberration, showing telescope pointing direction without aberration

The International Celestial Reference Frame

To measure radio source positions and parallaxes accurately, reference sources with very accurately

measured positions must be available. To create this reference frame, radio telescope networks

have measured, and continue to measure, the positions of hundreds of quasars across the sky. This

is called the International Celestial Reference Frame (ICRF), shown in Figure 16.

The HartRAO 26m telescope has played a key role in this project, being one of the few southern

radio telescopes equipped for this. All the telescopes observe the same object at the same time, in a

technique known as Very Long Baseline Interferometry (VLBI). Observations are made at two

widely spaced frequencies (2.3 and 8.4 GHz) in order to measure and correct for the (varying)

effects of the Earth's atmosphere and ionosphere.

The networked radio telescopes are able to measure source positions to better than one milli-

arcsecond, whereas ground-based optical telescopes can achieve about 100 milli-arcseconds. As a

result this extra precision, the radio source-based reference frame replaced the optical FK5

reference frame in 1998.

The ICRF is crucial for comparing the positions of objects observed at different wavelengths across

the electromagnetic spectrum, i.e. through multi-wavelength astronomy. In order to understand

the relationship between the emission seen at different wavelengths, it is necessary to accurately

align and overlay the images taken at each wavelength.

The Gaia spacecraft, to be launched in December 2011, is designed to set up a new optical

reference frame aiming at 24 micro-arcsecond accuracy, which will be aligned with the ICRF. The

radio reference frame itself will be improved by moving to higher frequencies (up to 32GHz) which

will provide higher precision.

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Figure 16: International Celestial Reference Frame comprises

The International Terrestrial Reference Frame – GNSS, SLR, DORIS

At the same time as radio source positions are being measured precisely in the ICRF, so the

positions of the participating radio telescopes are also determined with high precision. After taking

out short-term effects such as tides in the solid Earth, we are left with long term motions of the

radio telescopes (Figure 17), which measure the motion of the tectonic plates on which they are

located (Figure 18). The accurate absolute positions of the these telescopes means that systems that

measure relative positions can be co-located with them, and the absolute positions transferred to

these systems. They include the Global Navigation Satellite Systems (GNSS) such as the US

Global Positioning System (GPS), Russian GLONASS and European Galileo, Satellite Laser

Ranging (SLR) and Doppler Orbitography and Radiopositioning Integrated by Satellite

(DORIS). These four systems are located at Hartebeesthoek, making it a key station in the

International Terrestrial Reference Frame (ITRF). There are many scientific and practical

spinoffs from the precise position measuring that these systems carry out.

Question: What uses of high precision position measurement with GPS can you think of ?

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Figure 17: Changing distance between SA and USA

Figure 18: Radio telescope motion vectors